# Slow continued fractions, transducers, and the Serret theorem

## TL;DR

This paper explores the structure of continued fraction algorithms using transducers and group actions, providing bounds on tail-equivalence classes and extending the classical Serret theorem.

## Contribution

It offers a uniform framework for understanding the Serret theorem across various continued fraction algorithms via finite transducers and subgroup classifications.

## Key findings

- Finitely many subgroup possibilities for Gauss map branches.
- Partition of real numbers into finitely many tail-equivalence classes.
- Conditions under which the Serret theorem holds almost everywhere.

## Abstract

A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). This result was first formulated in Serret's Cours d'alg\`ebre sup\'erieure, so we'll refer to it as to the Serret theorem. Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. In this paper we show that there are finitely many possibilities for the subgroups Sigma of PGL(2,Z) generated by the branches of the Gauss maps in a large family of algorithms, and that each Sigma-equivalence class of reals is partitioned in finitely many tail-equivalence classes, whose number we bound. Our approach is through the finite-state transducers that relate Gauss maps to each other. They constitute opfibrations of the Schreier graphs of the groups, and their synchronizability ---which may or may not hold--- assures the a.e. validity of the Serret theorem.